# Is physics rigorous in the mathematical sense?

I am a student studying Mathematics with no prior knowledge of Physics whatsoever except for very simple equations. I would like to ask, due to my experience with Mathematics:

Is there a set of axioms to which it adheres? In Mathematics, we have given sets of axioms, and we build up equations from these sets.

How does one come up with seemingly simple equations that describe physical processes in nature? I mean, it's not like you can see an apple falling and intuitively come up with an equation for motion... Is there something to build up hypotheses from, and how are they proven, if the only way of verifying the truth is to do it experimentally? Is Physics rigorous?

• What is wrong with experiments? – Bernhard Nov 14 '12 at 6:37
• If we are going to build Physics from ground-up, then there should be a set of axioms that we should lay by, isn't it? Or else, we could be building Physics on unstable structures. There is a famous saying, whom I forgotten the author, by which he says that, the idea of a proof is to prove beyond doubt, your own argument. – A New Guy Nov 14 '12 at 8:38
• But how would you know what constitutes a good set of axioms without first analyzing empirical data gathered by careful experimentation? In fact, mathematicians are in pretty much the same predicament. The common axiom sets found in formal mathematics were chosen to maximize theorem-proving power/efficiency, not because they are somehow "self-evident" truths. – David H Nov 14 '12 at 8:52
• On the other hand, you might say physicists try to rigorously adhere to the principles and procedures of mathematical statistics when it comes to quantifying their uncertainty in the reliability of a particular result. – David H Nov 14 '12 at 8:56
• The ideal of rigor for science is the hypothetico-deductive method. You make a hypothesis, deduce its consequences, and then test them against reality. The mathematical formalization of this process would be something like the AIXI algorithm in computer science, which uses data to make causal models. Also see the whole field of statistics, and its methods for establishing the likelihood of a hypothesis. The difference between mathematics and physics is that in physics you use empirical data as an input. But you can still be rigorous in your methods. – Mitchell Porter Nov 14 '12 at 10:39

No, physics is not rigorous in the sense of mathematics. There are standards of rigor for experiments, but that is a different kind of thing entirely. That is not to say that physicists just wave their hands in their arguments [only sometimes ;) ], but rather that it does not come even close to a formal axiomatized foundation like in mathematics.

Here's an excerpt from R.Feynman's lecture The Relation of Mathematics and Physics, available on youtube, which is also present in his book, Character of Physical Law (Ch. 2):

There are two kinds of ways of looking at mathematics, which for the purposes of this lecture, I will call the the Babylonian tradition and the Greek tradition. In Babylonian schools in mathematics, the student would learn something by doing a large number of examples until he caught on to the general rule. Also, a large amount of geometry was known... and some degree of argument was available to go from one thing to another. ... But Euclid discovered that there was a way in which all the theorems of geometry could be ordered from a set of axioms that were particularly simple... The Babylonian attitude... is that you have to know all the various theorems and many of the connections in between, but you never really realized that it could all come up from a bunch of axioms... [E]ven in mathematics, you can start in different places. ... The mathematical tradition of today is to start with some particular ones which are chosen by some kind of convention to be axioms and then to build up the structure from there. ... The method of starting from axioms is not efficient in obtaining the theorems. ... In physics we need the Babylonian methods, and not the Euclidean or Greek method.

The rest of the lecture is also interesting and I recommend it. He goes on (with an example of deriving conservation of angular momentum from Newton's law of gravitation and having it generalized):

We can deduce (often) from one part of physics, like the law of gravitation, a principle which turns out to be much more valid than the derivation. This doesn't happen in mathematics, that the theorems come out in places where they're not supposed to be.

• I see, that is a very helpful answer Stan! I will check out the lecture. However, is there absolutely no way, in no area of Physics, can we apply the idea of Axiom-Definition-Speculation-Theorem-Proof structure of mathematics? – A New Guy Nov 14 '12 at 8:40
• There's a sliding scale; some theoretical physics treatments to come closer to the style of mathematics. For example, there are axiomatic treatments of the formalism of quantum mechanics. Generally these areas are also influenced by mathematicians (e.g., von Neumann). – Stan Liou Nov 14 '12 at 8:47
• I guess mathematical constant arising in Euclidean geometry, the number $\pi$ a human could in theory compute more digits of it entirely mentally even if they have total sensory deprivation and have never learned what any of its digits are whereas the fine structure constant, the dimensionless constant arising in nature, people need to use the Babylonian method to figure out the digits of. – Timothy Sep 22 '18 at 23:42

Physics is usually not rigorous. But there is a branch of physics, called mathematical physics, in which physics is treated with full mathematical rigor. There everything begins with formally stated assumptions (axioms) from which everything else is rigorously deduced.

In particular, there are fully rigorous treatments of phenomenological thermodynamics (see, e.g., my paper http://www.mat.univie.ac.at/~neum/ms/phenTherm.pdf), of classical mechanics, of fluid mechanics, and of quantum mechanics.

A possible set of axioms for quantum mechanics is given in my ''Postulates for the formal core of quantum mechanics'' from Chapter A4: The interpretation of quantum mechanics of my theoretical physics FAQ at
http://www.mat.univie.ac.at/~neum/physfaq/physics-faq.html
This chapter also contains a discussion of ''What is the meaning of axioms for physics?''. See also ''Why bother about rigor in physics?'' in Chapter C2: Some philosophy of physics.

Some parts of physics are rigorous in the sense of mathematics - i.e. they are treated as mathematics, with physically-motivated entities. And it is not that uncommon that for some papers to make it rigorously, explicitly stating assumptions and proving things.

However, most of the time physics in not mathematically rigorous. It stems from a few things:

1. Physics, typically, work the other way around than mathematics. That is, knowing some effects you try to figure out assumptions so the effects can be explained.

2. Physics is related to the real world. And many times it is tricky to relate a pure mathematical concept to (reasonably) objectively measurable quantities.

3. In physics, there is not much difference if you fail at predicting because of error in mathematics, or using unphysical assumptions.

4. Most of things in physics started as a hand waving arguments, which were mathematically dubious, but "they worked in most cases". In that way it was possible to explain or predict many phenomena. Their mathematical grounding often went later (a few years, decades or... is still an open problem); and more than often, in a utterly unpractical form for any physical calculations.

Think of things like Lebesgue integral (still to get the actual numerical value you need to do summation or Riemann integration), delta Dirac as a distribution (for physical calculations it is treated as "narrow enough" function), formalization of path integrals (well, not suitable for calculations), ...

How does one come up with seemingly simple equations that describe physical processes in nature?

It is a big question.

There are some answer in the spirit of emergency like:

"Physics is this part of the reality that is easily described with mathematics."

Or even more cynically (it's more-or-less a quotation, but I've forgotten the author):

"Physics is this part of the reality that can be approximated as coupled harmonic oscillators."

Also, it is highly recommendable to read a classical text on that issue: Eugene Wigner, The Unreasonable Effectiveness of Mathematics in the Natural Sciences.

Well, you generally have a few fundamental equations, and you try to build a theory upon them. For example, one can consider the gravity equation $F=\frac{Gm_1m_2}{r^2}$ to be a fundamental. How was it derived? It wasn't -- it was experimentally determined to a known accuracy. Newton's laws can be considered to be "axioms" as well, though the Lagrangian formulism of Physics is more pleasing to look at axiomatically (some very simple statements involving energy are taken as axioms, and the rest is pretty much mathematically derived).

More modern theories like the Standard Model assume entities behave according to some mathematical relation -- again, this is your axiom (I'm not too familiar with SM, so I'm not too clear about this).

In the end, the only way to check is via experiments. It's technically the same with mathematics-- you "test" your axioms against your logical facilities. The axioms are pretty simple and sometimes silly at first glance, so you never know that you're doing this. On the other hand, in Physics, the axioms are more complex and don't sound as "silly". The experiments required to verify/come up with these are correspondingly more complex, and they don't give a completely definite result -- we can only say that "equation X is valid to Y accuracy" or something like that.

If we call the fundamentals "axioms", then yes, Physics is rigorous. But these axioms are on pretty shaky ground themselves. For the past century we've been trying to tweak these axioms/postulates so that we get a system consistent with the real world.

Another way to look at it is that we just port over all axioms from Mathematics, postulate a few things (what a force is, what momentum is, etc), and everything else is a conjecture that's been verified to a certain degree. Like Goldbach's conjecture.

I think that the second interpretation is the one commonly accepted -- I've never seen anything labelled as "axioms of physics", though certain things like special relativity can be derived in an axiomatic fashion (start with space and time, then go on)

• Hi. I don't mind taking the gravity equation as an axiom. What I want to ask (perhaps different from what the OP would like to ask) is: Suppose we assume certain axioms to describe the universe and use them to derive a statement P. Is the derivation of P from the axioms rigorous ? This is like the question: Is physics axiomatic ? – Amr Dec 19 '13 at 22:06

Not, always .. many times they used un mathematical models like 'dimensional regularization' or use some 'curve fitting' for some properties in several branches of physics or use 'conjectures' so the things fit

Since you are a student of mathematics with little knowledge of physics, I strongly urge you to take a few courses in modern physics before you finish your mathematics education (General Relativity and Quantum Mechanics the very least). This is a must if you are specializing in geometry/topology.

Having said that, the "axioms" of the two disciplines are not the same kind, and therefore the "rigor" of proofs is different.

Physical "axioms" are those that are invariant in every experiment. E.g., the "axiom" of conservation of energy; or that the speed of light cannot be exceeded, and so on. Newton's laws were once "axioms" but they had to be modified in extreme conditions. But that does not mean Newton's laws are false: they are still true to the world we normally experience.

Mathematical axioms can be "abstract" but not based on any experiment: parallel lines do not intersect is an ancient axiom. Whether that is true or not in our universe is not important to a mathematician.

This answer of mine is relevant to this question also:

Ever since the time of Newton physics is about observing nature, quantifying observations with measurements and finding a mathematical model that not only describes/maps the measurements but, most important, it is predictive. To attain this, physics uses a rigorous self-consistent mathematical model, imposing extra postulates as axioms to relate the connection of measurements to the mathematics, thus picking a subset of the mathematical solutions for the model.

The mathematics is self-consistent and rigorous by the construction of a mathematical model. Its usefulness in physics is that it can predict new phenomena to be measured.

It is the demand for self-consistency that allows for falsification of a proposed mathematical model, by its predicting numbers found to be false.

The consistent euclidean model of flat earth is falsified on the globe of the earth, for example. This lead to spherical geometry as the model of the globe.

The whole research effort of validating the standard model at LHC, is in the hope that it will be falsified and open a window for new theories.

In this sense physics as a discipline is as rigorous as mathematics.

The processes of designing experiments and getting real numbers to test the models introduces errors and statistical distributions, which define the accuracy of the "rigor" expected when testing a model, so in this sense physics is not absolutely rigorous.

Pythagoras theorem was true once upon a time when the great philosopher discovered it. It is still true, and will always be, as far as Euclidean geometry is concerned. The laws of motion, too, were considered perfectly valid in the times of Newton. But, now we know that Relativity is far more general, and in a sense, much more truthful. So the basic difference lies here: mathematics, an abstract science built upon axioms laid down by former mathematicians, seems to describe reality in an elegant and all the more rigorous way; whereas, physics, a natural science built upon the axioms laid down by God himself, is itself a manifestation of the reality we live in. The difference lies in that, one is abstract, and the other real; but only scientists can understand why we cannot dispense with either of them, both being fundamental to the understanding of the world we live in.

There is only one axiom in physics: Nature is always right.

The most important corollary for that is: Physicists are almost always almost completely wrong.

The little bit of overlap between nature and physicists generates the electricity for your computer.